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Instrumental Variables Analysis with Weak Instruments 517

(assuming that the instrument is not weak). If, however, the instrument is not exogenous,
then, if the instrument is not weak, bn1TSLS ¡p b1 + cov(Zi, ui)>cov(Zi, Xi) b1. That is,
if the instrument is not exogenous, the TSLS estimator is inconsistent.

A p p e n di x

12.5 Instrumental Variables Analysis

with Weak Instruments

This appendix discusses some methods for instrumental variables analysis in the presence
of potentially weak instruments. The appendix focuses on the case of a single included
endogenous regressor [Equations (12.13) and (12.14)].

Testing for Weak Instruments

The rule of thumb in Key Concept 12.5 is that a first-stage F-statistic less than 10 indicates
that the instruments are weak. One motivation for this rule of thumb arises from an
approximate expression for the bias of the TSLS estimator. Let bO1 LS denote the probabil-
ity limit of the OLS estimator b1, and let b1OLS - b1 denote the asymptotic bias of the OLS
estimator (if the regressor is endogenous, then bn1 ¡p bO1 LS b1). It is possible to show
that, when there are many instruments, the bias of the TSLS is approximately
E(bnT1 SLS) - b1 ≈ (bO1 LS - b1)> 3E(F) - 14, where E(F) is the expectation of the first-
stage F-statistic. If E(F) = 10, then the bias of TSLS, relative to the bias of OLS, is approx-
imately 1/9, or just over 10%, which is small enough to be acceptable in many applications.
Replacing E(F) 7 10 with F 7 10 yields the rule of thumb in Key Concept 12.5.

The motivation in the previous paragraph involved an approximate formula for the
bias of the TSLS estimator when there are many instruments. In most applications,
however, the number of instruments, m, is small. Stock and Yogo (2005) provide a formal
test for weak instruments that avoids the approximation that m is large. In the Stock–
Yogo test, the null hypothesis is that the instruments are weak, and the alternative
hypothesis is that the instruments are strong, where strong instruments are defined to
be instruments for which the bias of the TSLS estimator is at most 10% of the bias of
the OLS estimator. The test entails comparing the first-stage F-statistic (for technical
reasons, the homoskedasticity-only version) to a critical value that depends on the number
of instruments. As it happens, for a test with a 5% significance level, this critical value ranges
between 9.08 and 11.52, so the rule of thumb of comparing F to 10 is a good approximation
to the Stock–Yogo test.